See for me this is something BOTH characters need to do. Diana more so because Clark has lived the everyman from childhood to adulthood. Diana is a Queen's daughter and would be afforded certain privileges that came with being royal etc and a demi goddess. I mentioned before I don't want Diana to be what she is not, so it is not that I am saying she needs to have an id to be compassionate or care etc because that is not true as we know. Diana's core is empathy. But I do want her to walk in man/womankind's shoes for a little time. Even if it is to take a sabbatical out of her very long life to see life as a regular person without being fawned over or have people in awe or with the trappings of royalty or diplomatic staff tending to her needs 24/7. Spirit of Truth did this perfectly for me. I am eager to see how Patty will do this in WW2. Now Clark being Clark as you say can achieve justice and embraced his identity and honor his heritage etc...that is true but the way DC is doing it is not good. There is a problem where he works and what he does and what he chooses to use as his platform. Ethics play a huge role and Clark, especially if he is doing investigative journalism. What BM v SM did was a fine example of Clark's hypocrisy regarding wanting to out and stop BM when all the while as SM himself he too is not exactly embraced by everyone...yet he using his power to influence public opinion on another man. I just though Clark had very little moral ground to stand on. And right now there is some very iffy stuff going down in Heroes in Crisis what Clark is allowing to pass as integrity because he of his personal life/relationships. Clark can do his job as a writer in a less hypocritical way but it involves not following certain troupes and DC does not seem able to do it. Why such a man isn't a foreign correspondent or travelling as freelance or a writer of novels etc I have no clue. Well I do...but that is DC's unwillingness to push the character into the 21st century and show what it means to be a man that actually likes writing because expressing ideas and using words challenges even the super in him and because it was a way to also change the world. The blog was a good idea but alas they backtracked. Clark's motivations to even become a journalist has been watered down. We were given stupid reasons in the DCEU. There is no proper motivation for it. When someone finds their passion and dream profession they will tell you exactly why and what made them love it and what drives them to do it...the idea they would opt for something because of someone they want to jump into bed with...that's not an admirable reason.
But back to being outed, I feel the new 52 was a great chance to do it in the multiverse and let us see that story of an outed Clark, Kal, Superman...which is essentially one man simply with different facets and see how he copes with Diana at his side of course. We have tons of stories with the dual identities. I want to see what Clark's world would be like and what he would do. Many normal men and women do cope with celebrity & power and having to deal with love and hate at the same time, even when their star is ascending or descending from movie stars, presidents, kings, writers, poets, artists etc etc...Superman has more than enough spine, gravitas, fortitude I think to cope with this. Cap certainly does. We could say a true test of a man as well is not only when he asks for no plaudits but when much is expected from him. I think there are so many great stories DC could do and so many variations and you could still end up with very good character development . Heck I wanted to see Clark and Diana just go off on space adventures because she is the most believable ally and comrade that could endure the pace and has the training and experience and being who she is she brings something unique. The wider DCU surely would have Gods and magic too. I really hope this new restructuring we can get some multiverse stories.